Find the general solution of the given differential equation.y'''-3y”+3y'-y=0

Start with the characteristic equationr3−3r2+3r−1=0Factor the constant. -1=(1*-1). First try r=1(1)3−3(1)2+3(1)−1=0True statement, so r-1=0 is a factorRearrange the terms so it's easier to take out the known factor.r3−r2−2r2+2r+r−1=0Factor out the (r-1) termr2(r−1)−2r(r−1)+(r−1)=0(r−1)(r2−2r+1)=0Factor the binomial(r-1)((r-1)(r-1))=0(r-1)(r-1)(r-1)=0Triple root, multiply by t for each repeatResult:Y=c1et+c2tet+c3t2et

Check the solution to "Find the general solution of the given differential..."

High School
Calculus and Analysis
Calculus 2
Differential equations

Arnav Heath

Open question

2022-08-19

Find the general solution of the given differential equation.y'''-3y”+3y'-y=0

Answer & Explanation

eggma84rd

Beginner2022-08-20Added 11 answers

Start with the characteristic equation

r^{3} - 3 r^{2} + 3 r - 1 = 0

Factor the constant. -1=(1*-1). First try r=1

{(1)}^{3} - 3 {(1)}^{2} + 3 (1) - 1 = 0

True statement, so r-1=0 is a factor
Rearrange the terms so it's easier to take out the known factor.

r^{3} - r^{2} - 2 r^{2} + 2 r + r - 1 = 0

Factor out the (r-1) term

r^{2} (r - 1) - 2 r (r - 1) + (r - 1) = 0

(r - 1) (r^{2} - 2 r + 1) = 0

Factor the binomial
(r-1)((r-1)(r-1))=0
(r-1)(r-1)(r-1)=0
Triple root, multiply by t for each repeat
Result:

Y = c_{1} e^{t} + c_{2} t e^{t} + c_{3} t^{2} e^{t}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 2

What is the derivative of the work function?
How to use implicit differentiation to find $\frac{d y}{d x}$ given $3 x^{2} + 3 y^{2} = 2$ ?
How to differentiate $y = \log x^{2}$ ?
The solution of a differential equation y′′+3y′+2y=0 is of the form
A) $c_{1} e^{x} + c_{2} e^{2 x}$
B) $c_{1} e^{- x} + c_{2} e^{3 x}$
C) $c_{1} e^{- x} + c_{2} e^{- 2 x}$
D) $c_{1} e^{- 2 x} + c_{2} 2^{- x}$
How to find instantaneous velocity from a position vs. time graph?
How to implicitly differentiate $\sqrt{x y} = x - 2 y$ ?
What is 2xy differentiated implicitly?
How to find the sum of the infinite geometric series given $1 + \frac{2}{3} + \frac{4}{9} + ...$ ?
Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?
A. 4.2
B. 4.4
C. 4.7
D. 5.1
What is the derivative of $\frac{x + 1}{y}$ ?
How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?
How to find the volume of a cone using an integral?
What is the surface area of the solid created by revolving $f (x) = e^{2 - x}, x \in [1, 2]$ around the x axis?
How to differentiate $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 4$ ?
The differential coefficient of $\sec (\tan^{- 1} (x))$ .

Didn't find what you were looking for?